The theorem states that every Goodstein sequence (see below) eventually terminates at 0.
Table of contents 
2 Examples of Goodstein sequences 3 Proof 
Elements of a Goodstein sequence appear to increase rapidly. For example, G(4) starts as follows:
Hereditary notation  Value 

2^{2}  4 
2·3^{2} + 2·3 + 2  26 
2·4^{2} + 2·4 + 1  41 
2·5^{2} + 2·5  60 
2·6^{2} + 6 + 5  83 
2·7^{2} + 7 + 4  109 
...  
2·11^{2} + 11  253 
2·12^{2} + 11  299 
... 
Hereditary notation  Value 

2^{22}+2+1  19 
3^{33}+3  7625597484990 
4^{44}+3  approximately 1.3 × 10^{154} 
5^{55}+2  approximately 1.8 × 10^{2184} 
6^{66}+1  approximately 2.6 × 10^{36305} 
7^{77}  approximately 3.8 × 10^{695974} 
7·8^{7·87 +
7·86 +
7·85 +
7·84 +
7·83 +
7·82 +
7·8 + 7}
+ 7·8^{7·87 + 7·86 + 7·85 + 7·84 + 7·83 + 7·82 + 7·8 + 6} +
... + 7·8^{2} + 7·8 + 7

approximately 6 × 10^{15151335} 
7·9^{7·97 +
7·96 +
7·95 +
7·94 +
7·93 +
7·92 +
7·9 + 7}
+ 7·9^{7·97 + 7·96 + 7·95 + 7·94 + 7·93 + 7·92 + 7·9 + 6} +
... + 7·9^{2} + 7·9 + 6

approximately 4.3 × 10^{369693099} 
... 
Goodstein's theorem can be proved (using techniques outside Peano arithmetic, see below) as follows: Given a Goodstein sequence G(m), we will contruct a parallel sequence of ordinal numbers whose elements are no smaller than those in the given sequence. If the elements of the parallel sequence go to 0, the elements of the Goodstein sequence must also go to 0.
To construct the parallel sequence, take the hereditary base n representation of the (n1)th element of the Goodstein sequence, and replace every instance of n with the first infinite ordinal number ω. Addition, multiplication and exponentiation of ordinal numbers is well defined, and the resulting ordinal number clearly cannot be smaller than the original element.
The 'basechanging' operation of the Goodstein sequence does not change the element of the parallel sequence: replacing all the 4's in 4^(4^4)+4 with ω is the same as replacing all the 4's with 5's and then replacing all the 5's with ω. The 'subtracting 1' operation, however, corresponds to decreasing the infinite ordinal number in the parallel sequence; for example, ω^(ω^ω) + ω decreases to ω^(ω^ω) + 4 if the step above is performed. Because the ordinals are wellordered, there are no infinite strictly decreasing sequences of ordinals. Thus the parallel sequence must terminate at 0 after a finite number of steps. The Goodstein sequence, which is bounded above by the parallel sequence, must terminate at 0 also.